Chapter 15, Problem 15.50QP

(a) Calculate the percent ionization of a 0.20 M solution of the monoprotic acetylsalic acid (aspirin) for which K_a=3.0 × 10⁻⁴. (b) The pH of gastric juice in the stomach of certain individual is 1.00. After a few aspirin tablets have been swallowed, the concentration of acetylsalicylic acid in the stomach is 0.20 M. Calculate the percent ionization of the acid under these conditions. What effect does the nonionized acid has on the membranes lining the stomach? (Hint: See the Chemistry in Action essay “Antacids and the pH Balance in Your Stomach” in Section 15.12.)

Interpretation Introduction

Interpretation:

The percent ionization of the given $0.20M$ monoprotic acetylsalicylic acid solution has to be calculated.

Concept Information:

Acid ionization constant ${\text{K}}_{\text{a}}$:

Acids ionize in water.  Strong acids ionize completely whereas weak acids ionize to some limited extent.

The degree to which a weak acid ionizes depends on the concentration of the acid and the equilibrium constant for the ionization.

The ionization of a weak acid $\text{HA}$ can be given as follows,

${\text{HA}}_{(aq)}\to {\text{H}}^{+}{}_{(aq)}+{\text{A}}^{\text{-}}{}_{(aq)}$

The equilibrium expression for the above reaction is given below.

$K_a=\frac{[{\text{H}}^{\text{+}}{\text{][A}}^{\text{-}}\text{]}}{[\text{HA]}}$

Where,

$K_a$ is acid ionization constant,

$[{\text{H}}^{\text{+}}\text{]}$  is concentration of hydrogen ion

$[{\text{A}}^{\text{-}}\text{]}$  is concentration of acid anion

$[\text{HA]}$ is concentration of the acid

Percent ionization:

A quantitative measure of the degree of ionization is percent ionization.

For a weak, monoprotic acid $\text{HA}$ percent ionization can be calculated as follows,

$\text{percent}\text{ionization}=\frac{{\text{[H}}_{\text{3}}{\text{O}}^{\text{+}}\text{]}}{\text{[HA]}}\times 100\%$

Answer to Problem 15.50QP

The percent ionization of given $0.20M$ acetylsalicylic acid solution is $3.9\%$

Explanation of Solution

From the given concentrations of acetylsalicylic acid solution and its $K_a$ value, the hydrogen ion concentration can be found out.

Then the percent ionization can be calculated from the obtained hydrogen ion concentration as follows,

Construct an equilibrium table and express the equilibrium concentration of each species in terms of $x$

$\begin{array}{cccccc}& C_9{H}_8{O}_4{}_{\left(aq\right)}& \rightleftarrows & H_{\left(aq\right)}^{+}& +& C_9{H}_8{O}_4{}_{\left(aq\right)}^{-}\\ Initial\left(M\right)& 0.20& & 0& & 0\\ Change\left(M\right)& -x& & +x& & +x\\ Equilibrium\left(M\right)& \left(0.20-x\right)& & x& & x\end{array}$

The $K_a$ of acetylsalicylic acid is $3.0\times {10}^{-4}$

$\begin{array}{c}{\text{K}}_{\text{a}}\text{=}\frac{{\text{[H}}^{\text{+}}{\text{][C}}_9{\text{H}}_7{\text{O}}_4^{-}\text{]}}{\left[{\text{C}}_9{\text{H}}_7{\text{O}}_4^{}\right]}\\ 3.0\times {10}^{-4}\text{=}\frac{{\text{x}}^{\text{2}}}{\text{(0}\text{.20-x)}}\end{array}$

Assuming that x is small compared to $0.20$ , we neglect it in the denominator:

$\begin{array}{c}3.0\times {10}^{-4}\text{=}\frac{{\text{x}}^{\text{2}}}{\text{(0}\text{.20)}}\\ \text{x}\text{=}\sqrt{\text{(0}\text{.20)}\text{×}3.0\times {10}^{-4}}\\ \text{=}7.7\times {10}^{-3}\text{M}\\ \left[{\text{H}}^{\text{+}}\right]\text{=}7.7\times {10}^{-3}\text{M}\end{array}$

The percent ionization can be calculated as follows,

$\begin{array}{l}\text{percent}\text{ionization=}\frac{{\text{[H}}^{\text{+}}\text{]}}{\text{[HA]}}\text{×100}\text{\%}\\ \text{=}\frac{{\text{[H}}^{\text{+}}\text{]}}{\left[C_9{H}_8{O}_4\right]}\text{×100}\text{\%}\\ \text{=}\frac{7.7\times {10}^{-3}\text{M}}{\text{0}\text{.20}\text{M}}\text{×100}\text{\%}\\ \text{=}3.9\%\end{array}$

Therefore, the percent ionization of given $0.20M$ acetylsalicylic acid solution is $3.9\%$

Interpretation Introduction

Interpretation:

The percent ionization of acetylsalicylic acid solution under the given condition has to be calculated.

Concept Information:

Acid ionization constant ${\text{K}}_{\text{a}}$:

Acids ionize in water.  Strong acids ionize completely whereas weak acids ionize to some limited extent.

The degree to which a weak acid ionizes depends on the concentration of the acid and the equilibrium constant for the ionization.

The ionization of a weak acid $\text{HA}$ can be given as follows,

${\text{HA}}_{(aq)}\to {\text{H}}^{+}{}_{(aq)}+{\text{A}}^{\text{-}}{}_{(aq)}$

The equilibrium expression for the above reaction is given below.

$K_a=\frac{[{\text{H}}^{\text{+}}{\text{][A}}^{\text{-}}\text{]}}{[\text{HA]}}$

Where,

$K_a$ is acid ionization constant,

$[{\text{H}}^{\text{+}}\text{]}$  is concentration of hydrogen ion

$[{\text{A}}^{\text{-}}\text{]}$  is concentration of acid anion

$[\text{HA]}$ is concentration of the acid

Percent ionization:

A quantitative measure of the degree of ionization is percent ionization.

For a weak, monoprotic acid $\text{HA}$ percent ionization can be calculated as follows,

$\text{percent}\text{ionization}=\frac{{\text{[H}}_{\text{3}}{\text{O}}^{\text{+}}\text{]}}{\text{[HA]}}\times 100\%$

pH of a solution is the negative of the base $\text{-10}$ logarithm of the hydronium ion or hydrogen ion concentration.

$\text{pH}\text{=}\text{-log}\left[{\text{H}}^{\text{+}}\right]$

$\text{Concentration}\text{of}{\text{H}}^{\text{+}}\text{=}{\text{10}}^{\text{-pH}}$

Answer to Problem 15.50QP

The percent ionization of acetylsalicylic acid solution under given condition is $0.30\%$

Explanation of Solution

Given,

The pH of gastric juice in the stomach of a certain individual is $1.00$.after a few aspirin tablet have been swallowed, the concentration of aspirin in the stomach is $0.20M$ .

The concentration of $H^{+}$ at pH $1$ can be determined as,

$\begin{array}{c}\text{Concentration}\text{of}{\text{H}}^{\text{+}}\text{=}{\text{10}}^{\text{-pH}}\\ \text{=}{\text{10}}^{\text{-1}}\\ \text{=}\text{0}\text{.10}\text{M}\end{array}$

According to the Lechatelier’s principle, the position of equilibrium is shifted in the direction of the unionized acid.

Constructing an equilibrium table using the known equilibrium concentration of each species as follows,

$\begin{array}{cccccc}& C_9{H}_8{O}_4{}_{\left(aq\right)}& \rightleftarrows & H_{\left(aq\right)}^{+}& +& C_9{H}_8{O}_4{}_{\left(aq\right)}^{-}\\ Initial\left(M\right)& 0.20& & 0.10& & 0\\ Change\left(M\right)& -x& & +x& & +x\\ Equilibrium\left(M\right)& \left(0.20-x\right)& & 0.10+x& & x\end{array}$

The $K_a$ of acetylsalicylic acid is $3.0\times {10}^{-4}$

$\begin{array}{c}{\text{K}}_{\text{a}}\text{=}\frac{{\text{[H}}^{\text{+}}{\text{][C}}_9{\text{H}}_7{\text{O}}_4^{-}\text{]}}{\left[{\text{C}}_9{\text{H}}_7{\text{O}}_4^{}\right]}\\ 3.0\times {10}^{-4}\text{=}\frac{\text{x}\times \left(0.10+x\right)}{\text{(0}\text{.20-x)}}\end{array}$

Assuming that $\left(0.20-x\right)\approx 0.20and\left(0.10+x\right)\approx 0.10$

$\begin{array}{c}3.0\times {10}^{-4}\text{=}\frac{\text{x}\times 0.10}{\text{0}\text{.20}}\\ \text{x}=\frac{0.20\times \left(3.0\times {10}^{-4}\right)}{0.10}\\ =6.0\times {10}^{-4}\text{M}\\ \left[{\text{H}}^{\text{+}}\right]\text{=}6.0\times {10}^{-4}\text{M}\end{array}$

The percent ionization can be calculated as follows,

$\begin{array}{l}\text{percent}\text{ionization=}\frac{{\text{[H}}^{\text{+}}\text{]}}{\text{[HA]}}\text{×100}\text{\%}\\ \text{=}\frac{{\text{[H}}^{\text{+}}\text{]}}{\left[{\text{C}}_9{\text{H}}_7{\text{O}}_4^{}\right]}\text{×100}\text{\%}\\ \text{=}\frac{\text{6}\text{.0}\times {\text{10}}^{-4}}{\text{0}\text{.20}\text{M}}\text{×100}\text{\%}\\ =0.30\%\end{array}$

The percent ionization of acetylsalicylic acid solution under given condition is $0.30\%$